Recently, the optimal design by computer simulations are widely conducted. The optimal design by the computer simulations, which is often conducted, is the optimal design by the numerical calculation. For example, as illustrated in FIG. 1, the horizontal axis represents a cost, and the vertical axis represents a performance, and it is presumed that a value closer to the origin is preferable for both of them. Then, relationships between the cost and the performance are obtained by the computer simulations, and as illustrated in FIG. 1, each point corresponding to one relationship can be plotted on the plane or space, which is mapped by the cost and the performance. Because a point that the cost is lower and the performance is better is preferable in FIG. 1, a point closer to the origin is selected as an optimized point. However, because the results of the computer simulations are obtained as discrete points, it is unknown whether or not feasible points exist between the points.
On the other hand, the optimal design by the computer simulations also includes an optimization method by the computer algebra. In this method, the computer simulations are carried out for various input parameter values to calculate output evaluation indicator values for each case. Then, as illustrated in FIG. 2, an approximate expression “a” that approximately expresses a relationship between the input parameter and the output evaluation indicator is calculated, and then, the optimization by the computer algebra is conducted based on this approximate expression “a”. As a processing for this optimization, there is a case that an expression representing a relationship between the cost and the performance as illustrated in FIG. 3 is calculated from the obtained approximate expression and constraint conditions. However, conventionally, an error of the approximate expression is not considered, though the approximate expression is used.
Incidentally, as for the computer algebra, a technique, which is called a Quantifier Elimination (QE), is well-known. This technique is a technique for transforming an expression such as ∃x (x2+bx+c=0) to an equivalent expression in which a quantifier (∃ and ∀) is eliminated such as b2−4c≧0.
Specifically, the QE method is described in the following document. However, because a lot of documents for the QE method exist, useful documents other than the following document exist. This document is incorporated herein by reference.
Anai Hirokazu and Yokoyama Kazuhiro, “Introduction to Computational Real Algebraic Geometry”, Mathematics Seminar, Nippon-Hyoron-sha Co., Ltd., “Series No. 1”, Vol. 554, pp. 64-70, November, 2007, “Series No. 2”, Vol. 555, pp. 75-81, December, 2007, “Series No. 3”, Vol. 556, pp. 76-83, January, 2008, “Series No. 4”, Vol. 558, pp. 79-85, March, 2008, “Series No. 5”, Vol. 559, pp. 82-89, April, 2008.
Anai Hirokazu, Kaneko Junji, Yanami Hitoshi and Iwane Hidenao, “Design Technology Based on Symbolic Computation”, FUJITSU, Vol. 60, No. 5, pp. 514-521, September, 2009.
Jirstrand Mats, “Cylindrical Algebraic Decomposition—an Introduction”, Oct. 18, 1995.
Namely, there is no conventional technique for visualizing an error of a model to be processed in the computer algebra.